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I once asked if it is possible calculate a "direct" fix (from two altitudes) using Haversines only. Lars Bergman was kind to provide an example here (http://fer3.com/arc/m2.aspx/All-Haversine-Fix-Bergman-sep-2018-g42693).

I played with it - seems no alterations or amendments are required, it works just fine if three simple rules are followed:

N;E - positive, S;W - negative

p1 should be calculated off the Northern Pole

the "second" body (which GHA is used for Lon) should preferably be to the west (t2 to be positive), which will simplify the last step

The accuracy with the 4-place NatHav table is reasonable - mostly within 3-:-5 arcminutes off.